Complexity of parallel machine scheduling with processing-plus-wait due dates to minimize maximum absolute lateness

European Journal of Operational Research 114 (1999) 403±410

Theory and Methodology

Complexity of parallel machine scheduling with processing-pluswait due dates to minimize maximum absolute lateness T.C. Edwin Cheng a

a,*

, Mikhail Y. Kovalyov

b

Department of Management, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China b Institute of Engineering Cybernetics, National Academy of Sciences of Belarus, Surganova 6, 220012 Minsk, Belarus Received 1 March 1997; accepted 1 March 1998

Abstract We study the problem of scheduling n jobs on several parallel identical machines. An optimal combination of a job schedule and processing-plus-wait (PPW) due dates is to be determined so as to minimize the maximum absolute lateness. The problem is shown to be strongly NP-hard if the number of machines is variable and ordinary NP-hard if it is a constant greater than one. For the single machine case, the problem is shown to be solvable by a graphical approach in O…n log n† time. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Parallel machine scheduling; Due date assignment

1. Introduction In this paper, we study the parallel machine scheduling problem with processing-plus-wait (PPW) due dates to minimize the maximum absolute lateness, which may be stated as follows. There are n independent non-preemptive jobs to be scheduled for processing on m identical parallel machines. It is assumed that each machine handles jobs from time zero onwards without idle time and at most one job at a time. Each job j has an integer processing requirement pj > 0 and it is assigned a PPW due date dj ˆ kpj ‡ d, where k P 0

*

Corresponding author. Tel.: 852 2766 5216; fax: 852 2356 2682; e-mail: [email protected]

and d are decision variables which have to be determined. Due to the no idle time assumption, a schedule is completely characterized by the sequences of jobs on the machines. Given a schedule, the completion time Cj of each job j is easily determined. Given a schedule and values of k and d, the lateness of job j is de®ned as Lj ˆ Cj ÿ dj . The objective is to ®nd an optimal schedule and optimal values of k and d so as to minimize the maximum absolute lateness maxfjLj jg. Here and below we assume that each maximum and minimum is taken over all jobs j if not de®ned di€erently. Motivation of this problem stems from the construction industry. A contractor plans to bid for n construction projects (i.e. jobs) from a developer. The projects will be assigned to m teams of construction crew (i.e. machines) to handle. In

0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 8 ) 0 0 1 1 1 - 8

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T.C.E. Cheng, M.Y. Kovalyov / European Journal of Operational Research 114 (1999) 403±410

the bidding document, the contractor will have to specify the delivery date for each project with full justi®cation. An acceptable practice for estimating project delivery dates is to use the PPW due date assignment method. The contractor is keen to obtain the contracts and, based on past experience, he is sure to win the projects if he tenders a low enough bid for these projects. Hence, the revenue from the contracts is relatively ®xed and, in order to maximize pro®t, he has to minimize costs. There are costs incurred in the construction work which the contractor has tight control. There are costs which are beyond the contractor's full control. One of these costs is the cost due to deviation of the completion date from the delivery date for each project because there are signi®cant contractual penalties for missing the delivery date. To be conservative, the contractor tends to minimize the maximum deviation of completion date from delivery date among all projects. Kahlbacher [1] and Kahlbacher and Cheng [2] analyze various due date assignment methods for scheduling problems. They state that PPW due dates are superior for job-shop systems with dynamically arriving jobs and ``recommend to use PPW due dates for static single machine problems, if one conjectures that the problem instances ful®ll to a certain degree the properties of exponential distributed processing times, i.e. possess a high portion of rather short jobs and a decreasing portion of relatively large jobs'' [2]. For the case of a single machine, the above problem has been studied by Kahlbacher and Cheng [2]. They established some properties of an optimal solution for this problem and formulated it as a linear program with three variables and 2n constraints. The most closely related parallel machine problem has been studied by Li and Cheng [3]. In this problem, job weights and k ˆ 0 are given, i.e. there is a common due date d to be determined and the objective is to minimize the maximum weighted absolute lateness. Li and Cheng proved that this problem is ordinary NPhard for the single machine case and strongly NPhard for the general case and presented a polynomial time heuristic. The concept of PPW due dates was introduced by Smith and Seidmann [4]. Surveys of recent results on scheduling problems

with various models of due date assignment and earliness and tardiness penalties are provided by Cheng and Gupta [5] and Baker and Scudder [6]. The practical importance of assigning accurate due dates to jobs has been discussed in [7,8]. In this paper, we show that the single machine case of our problem is solvable in O…n log n† time and prove that the problem is ordinary NP-hard if the number of machines m P 2 is a constant and it is strongly NP-hard if m is a variable. The strong NP-hardness of the problem with a constant number of machines remains an open question. The paper concludes with some remarks and suggestions for further research. We notice that the no machine idle time assumption is one of the basic assumptions for single machine scheduling model (see Baker [9]). Without this assumption our problem becomes trivial. To solve the problem in this case, d can be set to zero and k can be chosen so large that the value of the smallest due date and the di€erence between any two distinct due dates is greater than the total of the job processing times. For each distinct processing time, jobs with equal processing times can easily be scheduled around their common due date so that the maximum absolute lateness is minimized. 2. Single machine We adopt the three-®eld notation for scheduling problems introduced by Graham et al. [10] to denote our family of problems as follows: a=dj ˆ kpj ‡ d= maxfjLj jg. The ®rst ®eld in this notation, a 2 f1; Pm; P g, denotes the machine environment. If a ˆ 1, then there is a single machine. If a ˆ Pm, then there are m machines and m is a constant. If a ˆ P , then there are again m machines but m is a variable. The second ®eld indicates that PPW due dates are assigned to the jobs and the third ®eld refers to the objective function to be minimized. The general problem is represented by P =dj ˆ kpj ‡ d= maxfjLj jg. In this section, we show how to solve the single machine problem, 1=dj ˆ kpj ‡ d= maxfjLj jg, in O…n log n† time. Let the Shortest Processing Time (SPT) job sequence be a sequence of the jobs in which they are

T.C.E. Cheng, M.Y. Kovalyov / European Journal of Operational Research 114 (1999) 403±410

in non-decreasing order of their processing times. The following theorem proved by Kahlbacher and Cheng [2] is used to provide the basis for our algorithm. Theorem 1 [2]. There exists an optimal solution for the problem 1=dj ˆ kpj ‡ d= maxfjLj jg in which k P 1 and the jobs are in the SPT sequence. We note that Theorem 1 only identi®es certain properties of an optimal solution for 1=dj ˆ kpj ‡ d= maxfjLj jg and does not yield a procedure to solve the problem. Our algorithm is based on a graphical presentation of the problem. Assume that k P 1 and jobs are numbered in the SPT order so that p1 6


6 pn . Given the sequence …1; . . . ; n†, the completion P time of job j is calculated as follows: Cj ˆ jiˆ1 pi for j ˆ 1; . . . ; n. Then the lateness Lj of job j may be expressed as a function of the variables k and d as follows: Lj …k; d† ˆ Cj ÿ kpj ÿ d. Let k  P 1 and d  be some optimal values of k and d for the problem 1=dj ˆ kpj ‡ d= maxfjLj jg and let F  be the corresponding optimal objective function value. Lemma 1. The value of k  minimizes f …k; 0† ˆ maxfLj …k; 0†g ÿ minfLj …k; 0†g: The value of d  can be found as d  ˆ …maxfLj …k  ; 0†g ‡ minfLj …k  ; 0†g†=2: The optimal objective function value is calculated as F  ˆ f …k  ; 0†=2. Proof. Consider solutions satisfying Theorem 1. We ®rst show that the values of k and d are optimal if and only if (i) minfLj …k; d†g ˆ ÿ maxfLj …k; d†g and (ii) f …k; d† ˆ maxfLj …k; d†g ÿ minfLj …k; d†g is minimized. Assume that k and d are optimal and condition (i) is not satis®ed. Then changing the value of d may reduce the value of maxfjLj …k; d†jg. Suppose now that (i) is satis®ed. We then have maxfjLj …k; d†jg ˆ f …k; d†=2;

405

i.e. k and d should minimize both these functions. Therefore, if (ii) is not satis®ed, then k and d are not optimal. Conversely, if (i) and (ii) are satis®ed, then k and d are optimal. It is easy to see that f …k; d† ˆ f …k; 0† for any k and d. Then k  is a solution for the problem of minimizing f …k; 0†, subject to k P 1. Having obtained the value of k  , the value of d  can be found from the following chain of equations d  ˆ maxfLj …k  ; 0†g ÿ maxfLj …k  ; d  †g ˆ maxfLj …k  ; 0†g ‡ minfLj …k  ; d  †g ˆ maxfLj …k  ; 0†g ‡ minfLj …k  ; 0†g ÿ d  : Thus, we have d  ˆ …maxfLj …k  ; 0†g ‡ minfLj …k  ; 0†g†=2: Using properties (i), (ii) and the equation f …k  ; d  † ˆ f …k  ; 0†, the optimal objective function value is calculated as F  ˆ f …k  ; 0†=2.  We now present a graphical approach to solving the problem of minimizing f …k; 0†. Recall that where f …k; 0† ˆ maxfLj …k; 0†g ÿ minfLj …k; 0†g, Lj …k; 0† ˆ Cj ÿ pj k is a linear decreasing function of the variable k for j ˆ 1; . . . ; n. We draw each function Lj …k; 0† as a straight line (Line j) in the kLj …k; 0† plane. It is easy to see that the functions maxfLj …k; 0†g and minfLj …k; 0†g correspond to the upper envelope (UE) and lower envelope (LE), respectively, for the family of lines f1; . . . ; ng. It is evident that the optimal value of k which minimizes the function f …k; 0† is reached at the point where the UE and LE are closest to each other. As an example to illustrate the graphical approach, we consider a 3-job problem with processing times p1 ˆ p2 ˆ 2; p3 ˆ 3. Hence, Line 1 is L1 …k; 0† ˆ 2 ÿ 2k, Line 2 is L2 …k; 0† ˆ 4 ÿ 2k and Line 3 is L3 …k; 0† ˆ 7 ÿ 3k. A plot of these three lines on the k ÿ Lj …k; 0† plane is shown in Fig. 1 from which the UE and LE are easily constructed. It follows that the optimal solutions are determined as: 3 6 k  6 5 and d  ˆ 3 ÿ 2k  with a minimum objective function value F  ˆ 1. It is well known from computational geometry that UE and LE can both be constructed in O…n log n† time. Assume that UE and LE are found and UE is represented by two sequences: the

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T.C.E. Cheng, M.Y. Kovalyov / European Journal of Operational Research 114 (1999) 403±410

Fig. 1. A plot of Lj …k; 0† versus k for the example.

sequence of lines …ur ; . . . ; u1 †, where pu1 <


< purÿ1 < pur , and the sequence of cross points …krÿ1 ; . . . ; k1 †, where ki is the cross point (the value of k) of the lines ui and ui‡1 . Since pu1 <


< purÿ1 < pur , we have krÿ1 < krÿ2 <


< k1 . Similarly, assume that the LE is designated by the sequence of lines …l1 ; . . . ; lt †, where pl1 < pl2 <


< plt , and the sequence of cross 0 †, where ki0 is the cross point of points …k10 ; . . . ; ktÿ1 the lines li and li‡1 . Having constructed the UE and LE, the value of k  may be found in O…n† time as follows. If the UE and LE are both made up of a single line, i.e. r ˆ t ˆ 1, then any k P 1 is optimal. If r > 1 or

t > 1, then we observe that k  2 fk1 ; . . . ; 0 g. To ®nd k  , we merge the sekrÿ1 ; k10 ; . . . ; ktÿ1 0 † to form a quences …k1 ; . . . ; krÿ1 † and …k10 ; . . . ; ktÿ1 sequence …a1 ; . . . ; ar‡tÿ2 † such that a1 6


6 ar‡tÿ2 . When merging, we calculate for each aj indices xj and yj of the lines from the UE and LE, respectively, which are crossed with the line k ˆ aj . Then the value of k  is found as k  ˆ as where Cxs ÿ Cys ÿ as …pxs ÿ pys † ˆ minfCxj ÿ Cyj ÿ aj …pxj ÿ pyj †jj ˆ 1; . . . ; r ‡ t ÿ 2g:

We now present a formal description of the merging algorithm. In this algorithm MER, the

T.C.E. Cheng, M.Y. Kovalyov / European Journal of Operational Research 114 (1999) 403±410

sequence of triples …aj ; xj ; yj † is constructed for j ˆ 1; . . . ; r ‡ t ÿ 2. We use counters X and Y to store the indices of the current lines from the UE and LE, respectively. Algorithm MER Input: sequences …ur ; . . . ; u1 † and …krÿ1 ; . . . ; k1 † for the UE, sequences …l1 ; . . . ; lt † and …k10 ; . . . ; 0 † for the LE. We have ur > . . . > u1 ; ktÿ1 0 . krÿ1 < . . . < k1 ; l1 < . . . < lt ; k10 < . . . < ktÿ1 Output: sequence …a1 ; x1 ; y1 †; . . . ; …ar‡tÿ2 ; xr‡tÿ2 ; yr‡tÿ2 †. Step 1: Set j ˆ 1; s ˆ r ÿ 1; v ˆ 1; X ˆ ur ; Y ˆ l1 ; k0 ˆ kt0 ˆ ‡1. Step 2: Calculate aj ˆ minfks ; kv0 g. If aj ˆ ks , then set xj ˆ us ; X ˆ xj ; yj ˆ Y and s ˆ s ÿ 1. If aj ˆ kv0 , then set yj ˆ lv‡1 ; Y ˆ yj ; xj ˆ X and v ˆ v ‡ 1. If s ˆ 0 and v ˆ t, then stop: sequence …a1 ; x1 ; y1 †; . . . ; …ar‡tÿ2 ; xr‡tÿ2 ; yr‡tÿ2 † is constructed. Otherwise, set j ˆ j ‡ 1 and repeat Step 2. Since r ‡ t 6 2n, the time complexity of the algorithm MER is O…n†. Thus, we have shown in this section that the single machine problem 1=dj ˆ kpj ‡ d= maxfjLj jg can be solved in O…n log n† time.

3. Parallel machines In this section, we study the general problem with parallel machines. We prove that the problem with a constant number of machines, Pm=dj ˆ kpj ‡ d= maxfjLj jg, is ordinary NP-hard if m P 2 and that the problem with a variable number of machines, P =dj ˆ kpj ‡ d= maxfjLj jg, is strongly NP-hard. Prior to the NP-hardness proofs we establish some properties of an optimal solution for the parallel machine problem similar to those given by Kahlbacher and Cheng [2] for the single machine case.

407

Theorem 2. There exists an optimal solution for the problem P =dj ˆ kpj ‡ d= maxfjLj jg in which k P 1 and the jobs on each machine are sequenced in the SPT order. Proof. We ®rst prove that there exists a solution with k P 1. Consider an optimal solution with 0 6 k 6 1 and we will show that setting k ˆ 1 does not a€ect the optimality. Let …il1 ; . . . ; ilrl † be the sequence of jobs on machine l in the optimal solution. We have Cil

j‡1

ˆ Cilj ‡ pil

j‡1

and Cilj ÿ d 6 Cilj ÿ d ‡ …1 ÿ k†pil ‡ kpilj j‡1

from where it follows Lilj ˆ Cilj ÿ kpilj ÿ d 6 Cil ÿ kpil ÿ d ˆ Lil : j‡1

j‡1

j‡1

Hence, the lateness is ordered Lil 6 Lil 6


6 Lilr 1

2

l

on each machine and the objective value is equal to max fjLil j; jLilr jg:

16l6m

1

l

This value is minimized only if min fLil g :ˆ Lia1 ˆ ÿ max fLilr g :ˆ ÿLibr :

16l6m

16l6m

1

l

b

The latter equation implies Cia1 ÿ kpia1 ÿ d ˆ ÿCibr ‡ kpibr ‡ d: b

b

Since Cia1 ˆ pia1 , we have d ˆ …pia1 …1 ÿ k† ÿ kpibr ‡ Cibr †=2: b

b

The objective value can be written as Libr ˆ Cibr ÿ kpibr ÿ d b

b

b

ˆ …Cibr ÿ …pia1 ‡ k…pibr ÿ pia1 ††=2: b

b

If pibr P pia1 , then it is optimal to set k ˆ 1: b Otherwise, k ˆ 0 and the optimal objective value is equal to …Cibr ÿ pia1 †=2 :ˆ F : Consider this situab tion. Let us reverse the order of jobs on each machine. Denote the completion time and the

408

T.C.E. Cheng, M.Y. Kovalyov / European Journal of Operational Research 114 (1999) 403±410

lateness of job j in the new schedule by Cj0 and L0j , respectively. We have Ci0l ˆ Cilr ÿ Cilj ‡ pilj : l

j

Set k ˆ 1 and d ˆ F : Then L0il ˆ Cilr ÿ Cilj ÿ F : l

j

There are two cases to consider. 1. Cilr ÿ Cilj P F : Since Cilr ÿ Cilj 6 2F ; l

l

0 6 L0il j

6 Cilr ÿ Cilj 6 F : l

2. Cilr ÿ Cilj < F : Since Cilr P Cilj ; l

l

0 > L0il P ÿ F : j

We deduce that jL0j j 6 F for all jobs j, i.e. the new schedule, together with k ˆ 1 and d ˆ F , determine an optimal solution. Assume k P 1: The existence of an optimal solution in which the jobs are sequenced in the SPT order on each machine can be proved by applying a pairwise interchange of neighboring jobs for each machine. This technique is used by Kahlbacher and Cheng [2] to prove a similar statement.  We proceed with the NP-hardness proofs. Theorem 3. The problem Pm=dj ˆ kpj ‡ d= maxfjLj jg is NP-hard for any m P 2. Proof. We show that the decision version of the problem Pm=dj ˆ kpj ‡ d= maxfjLj jg is NP-complete by a transformation from the NP-complete problem PA R T I T I O N (see [11]): given positive is there a set XP N ˆ integers a1 ; . . . ; ar , P f1; . . . ; rg such that a ˆ A, where j j2X j2N aj ˆ 2A? Assume, without loss of generality, that aj 6 A for all j. Otherwise, PA R T I T I O N has a trivial solution with an answer ``no'' to the above question. Given any instance of PA R T I T I O N , we construct the following instance of the problem Pm=dj ˆ kpj ‡ d= maxfjLj jg. There are r ‡ 3m ÿ 2 jobs (m is a constant) including r partition jobs with processing times pj ˆ aj , m ÿ 2 A-enforcer

jobs with equal processing times pj ˆ A and 2m 2A-enforcer jobs with equal processing times pj ˆ 2A. We Pshow that there exists a set X  N for which j2X aj ˆ A if and only if there is a solution for our problem with a value maxfjLj jg 6 A. P If there is a set X  N for which j2X aj ˆ A, then we put k ˆ 2; d ˆ 0, assign two 2A-enforcer jobs to be scheduled last on each machine, schedule jobs of the set X in the SPT order on the ®rst machine, jobs of the set N n X in the SPT order on the second machine and one A-enforcer job on each machine l ˆ 3; . . . ; m. For this solution, we have 0 6 Cj 6 A; 0 6 dj ˆ 2pj 6 A for the partition jobs with pj ˆ aj 6 A=2: Note that there can be at most one partition job with pj > A=2 on any of the ®rst two machines. According to the SPT order, such a job must be scheduled last among the partition jobs assigned to the same machine. Thus, Cj ˆ A; 0 6 dj 6 2A for partition jobs with A=2 < pj 6 A: Further, Cj ˆ A; dj ˆ 2A for the Aenforcer jobs and Cj 2 f3A; 5Ag; dj ˆ 4A for the 2A-enforcer jobs. Then the value of this solution is maxfjLj jg ˆ maxfjCj ÿ dj jg ˆ A. We now assume that there is a solution for our instance of the problem Pm=dj ˆ kpj ‡ d= maxfjLj jg with a value not exceeding A. Then there is a solution with the following properties: (1) Exactly two 2A-enforcer jobs are scheduled consecutively and in the same time interval on each machine. If there is a job scheduled between two 2A-enforcer jobs on a certain machine, then the di€erence between their completion times is greater than 2A: Since these jobs have equal due dates, the maximum absolute lateness will be greater than A: Hence, there are exactly two 2A-enforcer jobs scheduled consecutively on each machine. Pairs of these jobs must be scheduled in the same time interval because otherwise two of them will have the di€erence between their completion times greater than 2A: (2) All other jobs precede the 2A-enforcer jobs on each machine. This property follows from Theorem 1 which states that the jobs can be sequenced in the SPT order on each machine. Properties (1), (2) and the no idle time assumption ensure that the total processing time of

T.C.E. Cheng, M.Y. Kovalyov / European Journal of Operational Research 114 (1999) 403±410

the partition jobs and A-enforcer jobs on each machine should be the same. Since the total processing time of these jobs is equal to mA, a partition job and an A-enforcer job cannot be processed on the same machine. De®ne X as the set of the partition jobs scheduled on one P of the machines. P a ˆ Then we have j2X j j2X pj ˆ A, as required.  Theorem 4. The problem P =dj ˆ kpj ‡ d= maxfjLj jg is strongly NP-hard. Proof. Here we use a transformation from the strongly NP-complete problem 3-PA R T I T I O N [4]: given positive integers a1 ; . . . ; a3r and A such that A=4 P < aj < A=2 for j 2 N ˆ f1; . . . ; 3rg, and j2N aj ˆ Ar, is there a partition of P the set N into r disjoint sets X1 ; . . . ; Xr such that j2Xl aj ˆ A for l ˆ 1; . . . ; r? Given any instance of 3-PA R T I T I O N , we construct an instance of the problem P =dj ˆ kpj ‡ d= maxfjLj jg in which there are r machines and 5r jobs with processing times pj ˆ aj for the partition jobs j 2 N and pj ˆ 2A for the enforcer jobs j ˆ 3r ‡ 1; . . . ; 5r. We show that there exists a solution for 3-PA R T I T I O N if and only if there exists a solution for our problem with a value maxfjLj jg 6 A. If N can Pbe divided into r disjoint sets X1 ; . . . ; Xr such that j2Xl aj ˆ A for l ˆ 1; . . . ; r, then we set k ˆ 2; d ˆ 0 and schedule the partition jobs of the set Xl followed by two enforcer jobs on machine l for l ˆ 1; . . . ; r. For this solution, we have 0 6 Cj 6 A; 0 6 dj ˆ 2pj 6 A for the partition jobs and Cj 2 f3A; 5Ag; dj ˆ 4A for the enforcer jobs. Then the value of this solution is maxfjLj jg ˆ A. Assume that there is a solution for our instance of the problem P =dj ˆ kpj ‡ d= maxfjLj jg with a value not exceeding A. Applying the same argument as presented in Theorem 3, we see that there is a solution possessing properties (1) and (2) (see Theorem 3) where the enforcer jobs replace the 2Aenforcer jobs and the partition jobs replace all other jobs. Consequently, the total processing time of the partition jobs on each machine is the same. We de®ne Xl as the set of the partition jobs scheduled on machine l for l ˆ 1; . . . ; r. Since P P j2N pj ˆ j2N aj ˆ Ar, we obtain

409

X X aj ˆ pj ˆ A for l ˆ 1; . . . ; r; j2Xl

j2Xl

as required.



Thus, the only open question is the strong NPhardness of the problem with a constant number of machines. 4. Conclusions The problem of scheduling n jobs on several parallel identical machines has been studied. In this problem, an optimal combination of a job schedule and processing-plus-wait due dates is to be determined so as to minimize the maximum absolute lateness. For the single machine case, we have shown that the problem can be solved in O…n log n† time. The general problem is proved to be strongly NP-hard. It is ordinary NP-hard if the number of machines is a constant greater than one. Further research can be undertaken to construct ecient enumeration and approximation algorithms and to study more general models with explicit due date assignment penalties included in the objective function. Properties of an optimal solution established in Theorem 2 can be used in the development of such algorithms. Acknowledgements The work of Cheng was supported in part by The Hong Kong Polytechnic University under grant number 351-193-A3-014; the work of Kovalyov was supported in part by INTAS (Projects INTAS-93-257, INTAS-93-257-Ext and INTAS96-0820). References [1] H.G. Kahlbacher, Due date assignment methods and single machine scheduling, Presentation at the Third International Workshop on Project Management and Scheduling in Como, 1992. [2] H.G. Kahlbacher, T.C.E. Cheng, Processing plus wait due dates in single machine scheduling, Journal of Optimization Theory and Applications 85 (1995) 163±186.

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[3] C.-L. Li, T.C.E. Cheng, The parallel machine min±max weighted absolute lateness scheduling problem, Naval Research Logistics 41 (1994) 33±46. [4] M.L. Smith, A. Seidmann, Due-date selection procedures for job-shop simulation, Computers and Industrial Engineering 7 (1983) 199±207. [5] T.C.E. Cheng, M.C. Gupta, Survey of scheduling research involving due-date determination decisions, European Journal of Operational Research 38 (1989) 156±166. [6] K.R. Baker, G.D. Scudder, Sequencing with earliness and tardiness penalties: A review, Operations Research 38 (1990) 22±36. [7] S.C. Wheelwright, Re¯ecting corporate strategy in manufacturing decisions, Business Horizons (1978) 57±66.

[8] G.L. Ragatz, V.A. Mabert, A framework for the study of due-date management in job-shops, International Journal of Production Research 22 (1984) 685±695. [9] K.R. Baker, Introduction to Sequencing and Scheduling, Wiley, New York, 1974. [10] R.L. Graham, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling, Annals of Discrete Mathematics 5 (1979) 287±326. [11] M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco, 1979.